Normal Distribution Percentile Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The percentile corresponding to a z-score is calculated with the cumulative distribution function Φ(z) of the standard normal distribution. A z-score of 1.96 corresponds to the 97.5th percentile: about 97.5% of values are below it.
Explanation
This calculator complements our z-score calculator: where that one converts a raw value into a z-score (the number of standard deviations separating it from the mean), this one does the next step by converting that z-score into a percentile — the proportion of values, in a normal distribution, that lie below that z-score. This conversion relies on the cumulative distribution function of the standard normal distribution, denoted Φ(z), which has no simple closed formula expressible with common elementary functions (it is expressed with the mathematical error function, erf). This calculator therefore uses a recognized numerical approximation, that of Zelen and Severo (1964), whose precision reported in the statistical literature is on the order of 7.5×10⁻⁸ — more than enough for practical use, and verified here against several known reference values from normal distribution tables before publication. The z-score of 1.96 is particularly well known in inferential statistics: it corresponds to the threshold used for a 95% (two-sided) confidence interval, already verified on our confidence interval calculator. A percentile of 97.5% for this z-score means that, in a normal distribution, 97.5% of values lie below it, leaving 2.5% beyond — and by symmetry, 2.5% below −1.96, which leaves 95% of values between −1.96 and 1.96.
Example: z = 1.96
Inputs
Z-score: 1.96.
Calculation
Φ(1.96) ≈ 0.975, that is a percentile of 97.5%.
Result
A z-score of 1.96 corresponds to the 97.5th percentile of the standard normal distribution.
Frequently asked questions
Why not calculate the normal cumulative distribution function with an exact formula?
Because the cumulative distribution function of the normal distribution has no closed formula expressible with standard elementary functions (addition, multiplication, exponential, square root): it is expressed with the mathematical error function (erf), itself defined by an integral with no simple algebraic solution. Highly accurate numerical approximations like Zelen and Severo's are therefore the standard method for this calculation in practice.
What does a percentile of 50% mean for a z-score of 0?
A z-score of 0 corresponds exactly to the mean of the distribution. In a normal distribution, perfectly symmetric about its mean, exactly half the values lie below the mean and the other half above — hence a percentile of 50% for a zero z-score, the central point of the distribution.
Does this calculator work for very extreme z-scores?
The approximation stays very accurate over the range of z-scores common in practice (generally between −4 and 4, which already covers more than 99.99% of usual cases). This calculator limits input to the range −6 to 6, more than enough for the vast majority of usual statistical applications, beyond which the probabilities involved become so tiny that they rarely have a direct practical use.